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The other day, while I was doing a bit of research into the Riemann Zeta function, I discovered a few interesting closed-form values of summations involving it, like $$\sum_{s=2}^\infty (\zeta(s)-1)=1$$ $$\sum_{s=1}^\infty (\zeta(2s)-1)=\frac{3}{4}$$ and when I checked Wikipedia, I found a ton of other more complicated sums involving the Zeta Function, like $$\sum_{s=2}^\infty \frac{\zeta(s)-1}{s}=1-\gamma$$ I know the trick that can be used for these sums is switching the order of the sums and using the fact that a geometric series is formed, but it fascinated me nevertheless.

QUESTION: Does anybody know of any other interesting examples of sums (that don't telescope trivially) with each term using non-elementary functions but a final answer that is elementary, or expressible in terms of other widely-known mathematical constants (like $\pi$, $e$, or $\gamma$)?

Specifically, I am looking for interesting identities regarding:

  • sums involving the inverses of functions that don't have elementary inverses, like the Lambert-W function, or the inverse of the function $f(x)=x+\sin x$
  • sums involving $\Gamma(s)$, at non-integer values of $s$
  • interesting generating functions for sequences that currently have no discovered closed-form explicit formula, like the Bell, Bernoulli, or Harmonic numbers
  • sums involving $\operatorname{Si}(s)$, $\operatorname{Li}(s)$, or $\operatorname{Ei}(s)$, or the elliptic integrals

Thanks!

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    $\begingroup$ Maybe also you are interested in Gregory coefficients, see this Wikipedia. I believe that Gregory was contemporary with Newton. Good luck. $\endgroup$ – user243301 Sep 11 '17 at 15:42
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    $\begingroup$ @user243301 : Good post, I think it fits well to the topic. :-) $\endgroup$ – user90369 Sep 11 '17 at 16:25
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    $\begingroup$ @user243301 Interesting, thanks! $\endgroup$ – Frpzzd Sep 12 '17 at 23:00
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In the non-exhaustive list below, you can find a lot of series involving special functions.

Among them many are interesting. For example, this one involving $\Gamma$ and $\zeta$ : $$\sum_{k=1}^\infty \frac{(-1)^k}{k!}\Gamma(k+x)\zeta(k+x)=-\Gamma(x)$$ Especially with $x=\frac{1}{2}$ one get a series representation for $-\sqrt{\pi}$.

It should be too long to edit many others from :

http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/23/01/ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/23/01/ http://functions.wolfram.com/Polynomials/HermiteH/23/02/ http://functions.wolfram.com/Polynomials/LaguerreL/23/02/ http://functions.wolfram.com/Polynomials/LegendreP/23/02/ http://functions.wolfram.com/Polynomials/ChebyshevT/23/01/ http://functions.wolfram.com/Polynomials/ChebyshevU/23/01/ http://functions.wolfram.com/Polynomials/GegenbauerC3/23/02/ http://functions.wolfram.com/Polynomials/JacobiP/23/01/ http://functions.wolfram.com/Polynomials/EulerE2/23/02/ http://functions.wolfram.com/Polynomials/BernoulliB2/23/02/ http://functions.wolfram.com/Polynomials/BellB2/23/02/ http://functions.wolfram.com/Polynomials/NorlundB2/23/02/ http://functions.wolfram.com/Polynomials/ZernikeR/23/01/ http://functions.wolfram.com/HypergeometricFunctions/ChebyshevTGeneral/23/01/ http://functions.wolfram.com/HypergeometricFunctions/GegenbauerC/23/01/

etc.

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    $\begingroup$ This is not a list of definitions. This is a list of sums. For example, if you open the link concerning the Hermite function, you will see 16 series. The first one is $$\sum_{n=0}^\infty \frac{H_n(z)w^n}{n!}=e^{2zw-w^2}$$ The series definitions of the special functions are not in the above list. Of course, it is easy to find them on the Wolfram's functions site. $\endgroup$ – JJacquelin Sep 11 '17 at 20:03
  • $\begingroup$ Yes, that's what I meant with includes a lot of formulas, that's good (!) to show the OP, that there are endless many examples. But e.g. the link of BernoulliB contents only a possible definition. :-) The OP doesn't say anything more about what he likes to know but I speculate that he expects some exotic series which cannot be find easily. $\endgroup$ – user90369 Sep 12 '17 at 5:19
  • $\begingroup$ OK. From the incomplete list of references, one can extract some series very difficult to find by ourselves as it is probably expected by the OP. I have not carried out a screening : it should be a tiresome task, not useful until the question be less wide. $\endgroup$ – JJacquelin Sep 12 '17 at 6:55
  • $\begingroup$ I agree. Till the OP doesn't specify more we have endless many possibilities for an answer. :-) $\endgroup$ – user90369 Sep 12 '17 at 9:44
  • $\begingroup$ Thus, it is up to the OP to chose among the references and examples provided in the answers what is the most useful for his purpose. I think that it is not possible for us to help him more. $\endgroup$ – JJacquelin Sep 12 '17 at 10:06

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